I have been learning a few new tricks about phenomenology of dark matter recently. Particularly, I started reading about inelastic dark matter. I have had the benefit to be able to talk to one of the originators of some of these ideas and I thought I would share some of them. Not because I believe that this is the correct solution to dark matter, but rather for its pedagogical value. There are various things that I learn again many times over in various guises and some of the details regarding this particular solution of the dark matter problem have that quality to them.
The first thing to know is that there is dark matter out there. It is supposed to be a relic of the big bang. There is lots of it out there (much more than ordinary matter). Current cosmological scenarios favor dark matter with weak scale annihilation cross sections and weak mass scales, suggesting that the physics of electroweak symmetry breaking might be closely related to the dark matter sector and therefore that we might produce dark matter candidates at the LHC.
The vanilla version of dark matter is that it consists of a single particle (called WIMP) that can interact by elastic collisions with ordinary matter and with cross sections that can be of order the weak scale. It is usually a fermion, but it can also be a boson.
The WIMP particle is heavy () and cold (moves slowly in the galaxy). Because the solar system is moving with respect to the Galaxy at about 200 Km/s, we should expect occasional recoils of matter with the WIMPs of about 200Km/s. This is the type of experiment being carried out in various places (DAMA, CDMS, …)
However, these experiments seem to contradict each other. DAMA sees a signal, and CDMS does not see a signal, so maybe there is a problem with the vanilla version of dark matter. To understand what could have happened, one finds out that the materials that make the DAMA detector are different to those that make the CDMS detector, so there might be some effect due to the difference in atomic mass of the materials that makes a difference when we consider these recoils.
For elastic collisions, one would not expect much difference. However, if collisions are inelastic, then when a WIMP interacts with matter it would produce a new particle (that is the definition of inelastic scattering). The new particle should be heavier than the WIMP and then energy conservation would forbid scattering unless the center of mass energy of the system is above the difference in mass between the new state and the old one. This energy must be kinetic energy.
So, assume we have a stationary nucleus, with mass , and a WIMP of mass moving at speed with respect to one another. How much kinetic energy do we have? This should be answered in the center of mass frame. That frame moves at speed
So the kinetic energy in the center of mass is
and this is equal to
So if the mass of the new particle is , we necessarily need to have
And then there is a dependence on the material. So if a material is lighter (like in CDMS) there might be not enough energy for the inelastic collision to take place.
The energy one needs is between 10KeV-100KeV depending on details in order to make the two experiments compatible. You should read the original paper if you want those details.
The tricky part is to think of an obvious mechanism that forces one to consider only inelastic collisions. This is where there are some fun observations on field theory that are appropriate. It turns out that a spin one particle can not decay into two identical spin zero particles.
A spin one particle is characterized by a polarization vector , and some four momentum , and they are required to satisfy . This means in the center of mass frame, you have angular momentum and only three independent polarizations. If one decays into two identical particles, the amplitude needs to be proportional to , so we need some other vector to contract it with. The only other vector available is the relative momenta between the decay products . So the amplitude is proportional to . But this means that the wave function of the identical particles is odd when we exchange the particles . However, this is incompatible with the statistics for bosons (the wave function needs to be symmetric in their exchange).
We can also understand this from effective field theory. We would need a vertex of the form
but after integration by parts we would find that this is
and $\partial_\mu B^\mu=0$ for a massive spin one particle (if we count only on-shell degrees of freedom). A more precise statement shows that this coupling can not happen due to gauge invariance of fundamental interactions. So in order to have the right type of coupling, the vector field needs to couple to two independent field theory degrees of freedom, that can have a mass splitting. That is the usual current for a symmetry.
What is the vector of choice? It needs to be neutral in order for us not to see extra charged particles in other experiments. In the original inelastic dark matter paper the vector of choice that mediated these collisions is the Z boson (coupling to the photon would tell us that the dark matter is charged and this is ruled out by observations). Of course, there are variants with many extra particles beyond the standard model.
The reason this is interesting, is that many times we make assumptions about the nature of interactions between particles without remembering some of the basics: many times interactions are mediated by particles of definite spin, and this can change the outcome of experimental signatures substantially.